\section{Greeks}
Often we want to know the sensitivity of the price of the claim with respect
to the price of the underlying asset, time, volatility or interest rate. These
sensitivities are the derivations of the function $V(S_t,t)$ and are called
the Greeks. Within the finite difference method they have the following
quantities:
\begin{eqnarray*}
\Theta&=\frac{dV}{dt}(i\Delta S_t,n\Delta t)&\approx\frac{V_i^{n+1}-V_i^n}{\Delta t}\\
\Delta&=\frac{dV}{dS_t}(i\Delta S_t,n\Delta t)&\approx\frac{V_{i+1}^{n}-V_{i-1}^n}{2\Delta S_t}\\
\Gamma&=\frac{d^2V}{d S_t^2}(i\Delta S_t,n\Delta t)&\approx\frac{V_{i+1}^{n}-2V_i^n+V_{i-1}^n}{\Delta S_t^2}\\
\rho&=\frac{dV}{dr}(i\Delta S_t,n\Delta t)&\approx\frac{V_{r+\Delta r}-V_{r-\Delta r}}{2\Delta r}\\
\nu&=\frac{dV}{d\sigma}(i\Delta S_t,n\Delta t)&\approx\frac{V_{\sigma+\Delta \sigma}-V_{\sigma-\Delta \sigma}}{2\Delta \sigma}\\
\end{eqnarray*}
The Black-Scholes PDE, equation \ref{eq:BSPDE}, can be rewritten in terms of
the Greeks resulting in
\[
\Theta + rS \Delta+ \frac{1}{2}\sigma^2S^2\Gamma=rV.
\]
For a Delta-neutral portfolio (risk neutral), $\Delta=0$ so
\[
\Theta + \frac{1}{2}\sigma^2S^2\Gamma=rV
\]
with $V$ the value of an option. This shows that when $\Theta$ is large and
positive, $\Gamma$ tends to be large and negative and vice versa. Therefore
$\Theta$ is useful to see the size (and sign) of $\Gamma$.\\
\\
\noindent
Before calculating all the values of the Greeks we checked their sensibility
regarding the values of $X_{min}$ and $X_{max}$. This is done for the $\Delta$
and shown in figure \ref{fig:Deltasensi} and \ref{fig:Deltasensi2}.  For the
figures in which the $X_{max}$ is changed the $X_{min}=250$ and when the
$X_{min}$ is changed the $X_{max}=150$.  These figures show that when the
premium  of an European put option is close to the analytical value for the
same option, so when the values of the boundaries are `far enough', the value
of $\Delta$ gets more realistic. In these pictures it is clear that $\Delta$
reflects the rate in which the premium changed with respect to the
stock price.\\
In all Greeks there is an irregularity at the left boundary, this is due to
the boundary conditions of the Finite Difference Scheme and should be
ignored.\\
\begin{figure}[htbp]
\begin{center}
\caption{Influence of $X_{max}$ on $\Delta$}\label{fig:Deltasensi}
\includegraphics[width=5.5cm]{delta_put_100_xmax25}\includegraphics[width=5.5cm]{delta_put_100_xmax50}\includegraphics[width=5.5cm]{delta_put_100_xmax75}\\
\includegraphics[width=5.5cm]{delta_put_100_xmax100}\includegraphics[width=5.5cm]{delta_put_100_xmax125}\includegraphics[width=5.5cm]{delta_put_100_xmax150}\\
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\caption{Influence of $X_{min}$ on $\Delta$}\label{fig:Deltasensi2}
\includegraphics[width=5.5cm]{delta_put_100_xmax200}\includegraphics[width=5.5cm]{delta_put_100_xmin200} \includegraphics[width=5.5cm]{delta_put_100_xmin400}\\
\includegraphics[width=5.5cm]{delta_put_100_xmin600} \includegraphics[width=5.5cm]{delta_put_100_xmin800} \includegraphics[width=5.5cm]{delta_put_100_xmax200_xmin1000}\\
\end{center}
\end{figure}

All Greeks are shown for an European put option which is `in the money', `at
the money' and `out of the money' respectively\footnote{The option on an
underlying asset with initial price respectively 100, 110 and 120 has a strike
of 110, volatility of 0.3, a time to maturity of one year in a world with a
0.04 interest rate}. The result are shown in figures
\ref{fig:Theta}, \ref{fig:Delta}, \ref{fig:Gamma}, \ref{fig:rho} and
\ref{fig:Vega}.\\
\newline
\noindent
$\Theta$, figure \ref{fig:Theta}, is the Greek that reflects the sensitivity
if the option premium with respect to the passage of time. Theta is usually
negative for an option, because as the time to maturity decreases with all
else remaining  the same the option tends to become less valuable. Theta is
not directly useful as hedge parameter it makes no sense to hedge against the
effect of passage of time on an option portfolio. The statistic is useful as a
descriptive statistic for a portfolio, and as shown earlier the value reflect
the value of Gamma as well.\\
\begin{figure}[htbp]
\begin{center}
\caption{$\Theta$}\label{fig:Theta}
\includegraphics[width=5.5cm]{theta_put_100}
\includegraphics[width=5.5cm]{theta_put_110}
\includegraphics[width=5.5cm]{theta_put_120}
\end{center}
\end{figure}
\noindent

The Greek $\Delta$, figure \ref{fig:Delta}, reflects the rate of change of the
option price with respect to the price of the underlying asset. This Greek was
used before for Delta hedging. For an European call option the Delta is equal
to $N(d_1)$ while for the European put option it is equal to $N(d_1)-1$ with
$d_1 =\frac{log(S_0/K) + (r+\sigma^2/2)T}{\sigma\sqrt T}$ as shown in our
previous report. From these equations it follows that when Delta hedging is
used for a short position in a European call option a long position of
$N(d_1)$ shares should be kept. Or the other way around, a Delta hedge for a
long position in an European call option should result in keeping a short
position of $N(d_1)$ shares at any given time. The Delta of a European put
option is negative, which means a long position in a European put option
should be hedged with a long position in the underlying stock.
The $\Delta$ is ascending quicker in the option that is `out of the money'
then when the option is `in the money'. Which means that the option is quicker
worth less for an increasing underlying stock.\\
\begin{figure}[htbp]
\begin{center}
\caption{$\Delta$}\label{fig:Delta}
\includegraphics[width=5.5cm]{delta_put_100}
\includegraphics[width=5.5cm]{delta_put_110}
\includegraphics[width=5.5cm]{delta_put_120}
\end{center}
\end{figure}
\noindent

$\Gamma$, figure \ref{fig:Gamma}, reflects the acceleration of the premium
with respect to the stock price, or in other words it reflects the change of
the Delta with respect to the price of the underlying asset. If Gamma is
small, Delta changes slowly which makes Delta hedging often easier. While when
Gamma is large, in absolute terms, then Delta is highly sensitive to the price
of the underlying asset. When this is the case the portfolio that is Delta
hedged should be changed frequently over time. \\
The acceleration of the premium which is `in the money' is more `flattened
out' then the acceleration for the premium which is `out of money'. This
is because a option that is `in the money' is worth more and is more stable
than an option which is `out of the money'.\\
\begin{figure}[htbp]
\begin{center}
\caption{$\Gamma$}\label{fig:Gamma}
\includegraphics[width=5.5cm]{gamma_put_100}
\includegraphics[width=5.5cm]{gamma_put_110}
\includegraphics[width=5.5cm]{gamma_put_120}
\end{center}
\end{figure}
\noindent

The Greek $\rho$, figure \ref{fig:rho}, reflects the sensibility of the
premium with respect to the risk free interest rate. For both the European
call and put option it is very easy to calculate rho. $\rho(call) =
KTe^{-rT}\Phi(d_2)$ and $\rho(put) = -KTe^{-rT} \Phi(-d_2)$. Figure
\ref{fig:rho} shows that a higher value in the stock price results in a higher
value for rho.\\
\begin{figure}[htbp]
\begin{center}
\caption{$\rho$}\label{fig:rho}
\includegraphics[width=5.5cm]{rho_put_100}
\includegraphics[width=5.5cm]{rho_put_110}
\includegraphics[width=5.5cm]{rho_put_120}
\end{center}
\end{figure}
\noindent

The Greek $\nu$, figure \ref{fig:Vega}, reflects the sensibility of the
premium with respect to the volatility. Up to now this wasn't important
because we assumed the volatility to be constant but in practice the
volatility changes over time. If Vega is high in absolute terms, the premium
is very sensitive to small changes in volatility. From figure \ref{fig:Vega}
it is clear that the volatility is very sensitive when the stock price is
equal to the strike price. When a portfolio is Vega neutral it is protected
for changes in the volatility.\\
\begin{figure}[htbp]
\begin{center}
\caption{$\nu$}\label{fig:Vega}
\includegraphics[width=5.5cm]{vega_put_100}
\includegraphics[width=5.5cm]{vega_put_110}
\includegraphics[width=5.5cm]{vega_put_120}
\end{center}
\end{figure}
